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Differential Models of Production: Change in the Marginal Cost and the Multi-Product FirmPowerPoint Presentation

Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

Lecture XXVI

- Shares of Marginal Cost
- Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price

- Based on this definition, we define a Firsch price index for inputs as

- Completing the single output model

- Expanding the production function to a multiproduct technology

- Expanding the preceding proof
- Computing the first-order conditions

- Now we replicate some of the steps from the preceding lecture, allowing for multiple outputs.
- Taking the differential of the first-order condition with respect to each output

- Again note by the first-order condition
- Thus

- With

- Differentiating with respect to the input prices yields the same result as before

- Slightly changing the preceding derivation by differentiating the production function by a vector of output levels, holding prices and other outputs constant yields

- Multiplying through by γyields
- Using the tired first-order conditions

- With

- Differentiating the production function with respect to yields

- Collecting these equations:
- Differentiating the first-order conditions with respect to ln(z’)
- Differentiating the first-order conditions with respect to ln(p’)

- Differentiating the production function with respect to ln(z’)
- Differentiating the production function with respect to ln(p’)

- The extended form of the differential supply system is then.
- Starting with the total derivative of ln(q)
- Premultiplying by F

- Note by the results from Barten’s fundamental matrix

- θir is the share of the ith input in the marginal cost of the rth product.
- Summing this marginal cost over all inputs

- Defining the matrix

- Expanding the differential model further, we introduce quasi-fixed variables into the production set

- Following Livanis and Moss, the differential supply function for this specification becomes

- Starting with the input demand system, we add a random disturbance relying on the theory of rational random behavior (RRB, Theil 1975):